On $n$th roots of normal operators

TL;DR

This paper studies the properties of nth roots of normal operators, showing they satisfy various spectral and decomposability conditions, and provides criteria for such operators to be normal.

Contribution

It establishes that nth roots of normal operators satisfy key spectral properties and characterizes when these roots are normal based on dominance or class conditions.

Findings

Nth roots of normal operators satisfy property (β)ε.

They are decomposable and have specific quasi-nilpotent parts.

Conditions for these roots to be normal are identified.

Abstract

For $n$-normal operators $A$ [2, 4, 5], equivalently $n$-th roots $A$ of normal Hilbert space operators, both $A$ and $A^*$ satisfy the Bishop--Eschmeier--Putinar property $(\beta)_{\epsilon}$, $A$ is decomposable and the quasi-nilpotent part $H_0(A-\lambda)$ of $A$ satisfies $H_0(A-\lambda)^{-1}(0)=(A-\lambda)^{-1}(0)$ for every non-zero complex $\lambda$. $A$ satisfies every Weyl and Browder type theorem, and a sufficient condition for $A$ to be normal is that either $A$ is dominant or $A$ is a class ${\mathcal A}(1,1)$ operator.